The manager intends to create a bond fund by investing proportions of the $20
million in the different securities and has announced an investment goal of a high-
quality, medium-maturity portfolio. In particular, the fund’s average quality rating
should be at least 3.5, and its average maturity should be no shorter than 1.5 years and
no longer than 2.5 years.
The portfolio manager seeks to create a bond fund that offers the highest
expected return subject to the quality and maturity requirements given. To accom-
plish this goal, what proportion of the $20 million should she invest in each bond?In the investment problem, the analyst seeks to maximize the portfolio return
subject to various constraints. Constrained optimization problems of this sort
form the core of a distinct managerial field known as operations research
(O.R.). Indeed, we all enjoy the benefits of operations research applications in
our everyday lives, although we may not know it. Suppose you decide to take
your family to Disney World:^1Operations research will be your invisible companion, scheduling the crews and air-
craft, pricing the plane tickets and hotel rooms, even helping to design capacities
on the theme park rides. If you use Orbitz to book your flights, an O.R. engine sifts
among millions of options to find the cheapest fares. If you get directions from
MapQuest, another O.R. engine spits out the most direct route. If you ship sou-
venirs home, O.R. tells UPS which truck to put the packages on, exactly where on
the truck the packages should go to make them fastest to load and unload, and
what route the driver should follow to make his deliveries efficiently.All of these operations involve maximizing, or minimizing, subject to con-
straints. The most basic and important tool of operations research is linear pro-
gramming.
Linear programming (LP)is a method of formulating and solving decision
problems that involve explicit resource constraints. Analysts use the LP method
to solve problems such as the investment problem and a host of other decision
questions: How should a firm allocate its advertising expenditure among vari-
ous media? What quantities of two jointly manufactured goods should a firm
produce with a fixed amount of labor and inputs? How should a federal agency
allocate its limited budget between two competing safety programs? What quan-
tities of output should a consumer-products firm transport from each of its fac-
tories to each of its retail outlets to minimize transportation cost?
What do these problems have in common? First, all seek to find the best val-
ues of certain variables: the right advertising mix, the most profitable product
quantities, the appropriate budget allocation. These values, which the decision
maker controls, are decision variables.Second, each decision has an explicit
objective, be it maximum profit, minimum cost, or maximum number of lives708 Chapter 17 Linear Programming(^1) This account comes from V. Postrel, “Operation Everything,” Boston Globe, June 27, 2004, p. D1.
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